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Geometry – Chapter 8
CHAPTER 8
–
SUMMARY
DIFFERENT WAYS TO WRITE A RATIO
Suppose you have the ratio
a
/b
.
Below are all the different ways in which you could be given the information of the ratio of a/b .
All the expressions below represent the same ratio a/b .
a
/b
a:b
a to b
a is to b
a out of b
YET ANOTHER WAY TO WRITE RATIOS
A ratio such as a : b : c is really two ratios lumped together:
a:b:c
this is ratio a : b
this is ratio b : c
EXAMPLES:
1)
C
The ratio 8 : 7 : 5 is actually the two ratios:
8:7
and
7:5
2) In the figure to the right, the ratio of sides AB : BC : CA is
What is the ratio of AB : BC? What is the ratio of BC: CA ?
3:4:5
A
SOLUTION:
1
AB : BC : CA is
3 : 4 : 5  This shows that the ratio of AB : BC is 3 : 4 = 3/4
AB : BC : CA is
3:4:5
 This shows that the ratio of BC : CA is 4 : 5 = 4/5
B
Geometry – Chapter 8
IMPORTANT POINT ON RATIOS:

A ratio represents a simplified fraction.

If the fraction is not simplified, then it’s not a ratio yet.
EXAMPLE:
1) Say you want to know the ratio of prices of two mp3 players:
RCA brand and iPod-brand. The prices of each are:
RCA-brand: $40
iPod-brand: $100
2) The order in which the ratio is stated is IMPORTANT:
 if you want the ratio of prices of RCA to iPod, you write the ratio:
RCA is first in the words “RCA to iPOD” ratio, so it is the numerator
40
100
iPOD is second in the words “RCA to iPOD” ratio, so it is the denominator
 if you want the ratio of prices of iPod to RCA, you write the ratio:
Now, iPod is first in the words “iPOD to RCA”, so iPOD is the numerator of the ratio
100
40
Now, RCA is second in the words “iPOD to RCA”, so RCA is the denominator of the ratio
3) Remember: a ratio is a simplified fraction, so the ratios above must be simplified!
 Simplify the ratio of costs of RCA to iPod:
40
40
4 
2
100
=
100
=
10 
=
5
 Simplify the ratio of costs of iPod to RCA:
100
100
10 
5
40
2
=
40
=
4

=
This is your final answer for ratio of RCA to iPod
2
This is your final answer for ratio of iPod to RCA
Geometry – Chapter 8
HOW TO SIMPLIFY RATIOS:
1) Check if ratio has units. Are the ratio’s numerator (top of fraction) and denominator (bottom
of fraction) units different?
EXAMPLE:

3 ft
6 in
3 ft
6 in
“Units” means the measurement description (ft, inches, lb, sec, etc.).
In this ratio, units are NOT the same.
(NOTE: If the units are the same, skip to step 4 below.)
2) Determine which units (numerator’s or denominator’s) are smaller, and convert the BIGGER
units to the smaller unit. (You don’t need to do anything to the part of the ratio that already has
the smaller units).
Here, “ft” is the bigger unit so I convert it to the smaller unit, “inches”
3 ft
6 in
In this example, “in” is the smaller unit so I don’t need to change the denominator.
3 ft
6 in
=
3 ( 12 in )
6 in
=
36 in
6 in
3) Check if now you can cancel out units. Cancel them out.
36 in
=
6 in
36 in
6 in
=
36
6
4) Simplify the fraction you are left with.
36
6
3
=
36 
6 
=
6
1
This is your final answer.
Notice the answer is
6
1
and not just “6”, because a ratio is always TWO NUMBERS
(it is a comparison of two numbers).
Geometry – Chapter 8
RATIOS AND RECTANGLES
W = width
1) If the ratio of length to width in a rectangle is a : b,
THEN:

The actual length of the rectangle is ax
 (that’s “a” multiplied by “x” )

The actual width of the rectangle is bx
 (that’s “b” multiplied by “x” )

The actual perimeter “P” of this rectangle is:
L = length
P = 2 (a+b) x
EXAMPLE:
If a rectangle has a width to length ratio of 3 : 5 and a perimeter of 64, find the width and length
of the rectangle.
Solution:
 the actual width of the rectangle is 3x
 the actual length of the rectangle is 5x
 the perimeter of the rectangle is:
P = 2 (3+5) x
This number
was given
64 = 2(8)x
64 = 16 x
 16
 16
4=x
Now that we know the value of x , go back and plug in to find actual width and length
 the actual width of the rectangle is 3x = 3 (4 ) = 12 = width
 the actual length of the rectangle is 5x = 5 (4) = 20 = length
4
Final
answers
Geometry – Chapter 8
WHERE ARE RATIOS IN REAL LIFE?

The scale of a map or the scale of a model airplane are ratios!!!
Example: A map with a scale of 1 in = 200 ft tells us that the ratio
map distance
1 in
=
real distance
200 ft

Bulk sales prices at stores are a ratio!!!
Example: If donuts are sold at 9 donuts for $3.00, this tells us that the ratio
quantity
9 donuts
9
=
=
price
$3.00
3

Currency exchange rates are a ratio !!!
(note: currency is the money used in each particular
country)
Example: In the U.S., the currency is the dollar. In Bolivia, the currency is the boliviano.
These days, every U.S. dollar is worth 7 bolivianos . This tells us that the ratio
dollars
1
bolivianos
=
7
WHAT IS A PROPORTION?
A proportion is an equation that shows that two ratios are equal.
EXAMPLE:
Determine if the pair of ratios below is a proportion.
35
and
40
21
24
SOLUTION:
Is
35
40
=
21
?
Let’s find out.
24
Using calculator, we find that 35/40 = 0.875
Using calculator, we also find that
5
21
/24
= 0.875
The two ratios are in fact equal to
each other, so this pair of ratios is
proportional.
Geometry – Chapter 8
SOLVING PROPORTIONS
To solve proportions, remember to cross-multiply!
a
b
Multiply these two
together and
place here.
c
d
=
Multiply these two
together and
ad place here.
=
bc
EXAMPLE 1
3x - 8
6
Solve the proportion
3x – 8
=
6
=
2x
10
.
2x
10
6 * 2x = (3x-8)(10)
12x = 30x – 80
12x + 80 = 30x
80 = 30x – 12x
80 = 18x
80
/18
=x
simplify:
x = 80/
EXAMPLE 2
18 =
40
/9
Answer
For the rectangle ABCD shown, AB : BC is 3 : 8. Solve for x.
A
B
SOLUTION:
Given 
AB
width
=
=
BC
length
x
3
8
D
From the figure, we can also see that
So:
width
length
=
Solve this proportion by cross-multiplying:
3
8
=
6
x
48 = 3x
16 = x
6
Answer
width = 6
3
8
=
6
x
and
length = x.
6
C
Geometry – Chapter 8
SIMILAR POLYGONS

What’s so special about similar polygons?
1) Their corresponding (matching) angles are congruent (same).
2) All the ratios of the corresponding (matching) sides are congruent (same).

BOTH (1) and (2) above must be true in order to prove two polygons are similar.
EXAMPLE
ABCD ~ RSTU (the symbol ~ means “similar”)
C
S
B
also is
=
R
T
D
U
A
 The corresponding (matching) angles are the same. These are marked with red arcs.
 The ratios of all corresponding (matching) sides are equal. The corresponding sides are matched
by color in the figures above. The ratios of these corresponding sides are:
AB
RS
7
=
BC
ST
=
CD
TU
=
DA
UR
Geometry – Chapter 8
EXAMPLE
x
116
(y – 73)
The two polygons to the right are similar.
Find the values of x and y.
61
4
6
116
5
SOLUTION:
Since the polygons are similar, we know the ratios of all
pairs of corresponding sides are congruent (the same).
x
4
6
5
Write out the ratios of these corresponding sides, always using the same
polygon as numerator for all your ratios (you can choose which polygon
you want as numerator):
This time, choose the bigger polygon to always use as numerator:
Corresponding sides
are matched by color
Ratio for green sides:
6
4
Ratio for pink sides:
x
5
From definition of similar polygons, these two ratios should be the same, so we can write:
Solve by cross-multiplying:
6
4
=
x
5
6
4
=
x
5
4x = 6 * 5
4x = 30
x = 30/4
x =
15
/2
Answer
We still need to solve for y:
From the definition of similar polygons, all corresponding angles are congruent (the same).
(y-73) 116
61
116
The angles marked with green are corresponding so they are equal.
Therefore, the missing angle in the big polygon is:
(y-73)
116
61
The interior angles of all 4-sided polygons always add up to 360 , so:
(y – 73) + 116 + 61 + 90 = 360
y – 73 + 267 = 360
8
y = 166
Answer
Geometry – Chapter 8
SHORTCUTS FOR SIMILAR TRIANGLES
SHORTCUT # 1 : ANGLE – ANGLE (AA) SIMILARITY
If two of the angles in one triangle are congruent to two of the angles of
another triangle, then those two triangles are similar.
Just check to see if two angles in one of the triangles are the same as two
triangles in the other triangle. If they are, then the two triangles are similar.
SHORTCUT # 2 : SIDE – SIDE – SIDE (SSS) SIMILARITY
b
If the ratios of all three pairs of corresponding sides of two triangles are the
same, then those two triangles are similar.
a
c
q
This shortcut is great to use if no information is given on the angles.
p
To use this shortcut, first write out all the ratios of all the corresponding
sides:
Remember to choose one of the triangles
r
a
b
c
p
q
r
as your numerator for ALL your ratios!
Here we chose this one.
b
Then, use a calculator to check if all three
ratios are equal. If they are, then the two
triangles are similar.
a
c
p
(Corresponding
sides are
q
matched by
color.)
r
SHORTCUT # 3 : SIDE – ANGE – SIDE (SAS) SIMILARITY
FOLLOW CHECKLIST
If:
1) One angle of one triangle is congruent to (same as) another angle
of another triangle
s
a
b
AND
t
2) The ratios of the corresponding sides that include those angles
are also congruent (the same)
THEN those two triangles are similar!
Follow this checklist to check if the two triangles in the drawing above are similar:
(1) We see both triangles have a congruent angle.
 CHECK.
(2) We check if the ratios of the corresponding sides that include (or “make”) the angles are
congruent:
a
b
Use a calculator to check if both ratios are equal.
s
t
If they are, then  CHECK.
If both (1) and (2) above
9
 CHECK, then the two triangles are similar.
Geometry – Chapter 8
EXAMPLE FOR SHORTCUT # 1 : ANGLE – ANGLE (AA) SIMILARITY
49
Are these two triangles similar?
56
SOLUTION: There are two pairs of congruent angles between both triangles.
In other words, both triangles have a 49 angle and a 56 angle.
56
49
Therefore, by the AA similarity theorem, both triangles are similar.
EXAMPLE FOR SHORTCUT # 2 : SIDE – SIDE – SIDE (SSS) SIMILARITY
3
6
Are these two triangles similar?
SOLUTION: We only have information on the triangles’ sides, so we try
with the SSS similarity theorem.
12
2
4
We must find all the ratios of all the corresponding sides to see if they are all
congruent. USE SAME TRIANGLE AS NUMERATOR FOR ALL RATIOS!
8
side “6” of the orange  corresponds to the side “4” of the green
side “3”
Looking at the figures:

of the orange  corresponds to the side “2” of the green 
side “12” of the orange  corresponds to the side “7” of the green 
Therefore, the ratios are:
6
4
Using a calculator:
=1.5
3
2
=1.5
12
8
=1.5

All ratios are congruent (same).
Therefore, by SSS similarity theorem,
the two triangles are similar.
EXAMPLE FOR SHORTCUT # 3 : SIDE – ANGE – SIDE (SAS) SIMILARITY
FOLLOW CHECKLIST
6 50 10
9
50
1) Both triangles have a 50 angle.  CHECK.
2) The sides of the pink triangle that include the 50 angle años
15
Follow this checklist to check if the two triangles in the drawing above are similar:
(1) We see both triangles have a congruent angle.
 CHECK.
(2) We check if the ratios of the corresponding sides that include (or “make”) the angles are
congruent:
a
b
Use a calculator to check if both ratios are equal.
s
t
If they are, then  CHECK.
If both (1) and (2) above
10
 CHECK, then the two triangles are similar.
Geometry – Chapter 8
PARALLEL LINES IN TRIANGLES
TRIANGLE PROPORTIONALITY THEOREM
A line parallel to one side of a triangle will divide the other two sides of the
triangle proportionally.
Said in other words,
a
/b
if these two lines are parallel ……….that means that the ratio of
a
is congruent to (the same as)
b
the ratio of
t
s
/t
s
a
b
CONVERSE THEOREM 
=
s
t
This theorem also works in reverse!
Say you have a triangle with a line that intersects a first and second side of a triangle
(such as the blue line intersecting the triangle in the figure below).
That intersecting line cuts the first side of the triangle into segments a and b , and
cuts the second side of the triangle into segments s and t .
If:
the ratio of
a
/b
on the first side = the ratio of
s
/t
on the second side.........
First Side
a
.........then the intersecting line (the blue line)
b
Third Side
the third side of the triangle.
t
s
Second Side
11
is parallel to
Geometry – Chapter 8
PARALLEL LINES AND TRANSVERSALS
Three parallel lines intersecting two transversals will divide those transversals proportionally.
(Note: transversals are lines that cut across the three parallel lines),
Let’s explain this better with an example.
In the figure below, three parallel lines (in black) intersect two transversals (one in blue and one in green).
a
b
The parallel lines cut the blue transversal into segments a and b.
The parallel lines also cut the green transversal into segments g and h.
g
h
According to this theorem, since the three lines cutting the transversals
are parallel, then the following ratios are equal:
a
b
=
g
h
PROPORTIONS IN TRIANGLES WITH A BISECTOR RAY
Say a ray bisects (splits in half) an angle of a triangle and splits the side opposite that angle into two
segments, a and b, as show in the figure below. The other sides of the triangle that are NOT intersected
by the ray are labeled sides f and g .
f
a
b
=
The ratio of the segments a and b
a
b
=
g
the ratio of the sides not intersected by the ray, f and g
f
g
NOTICE THE ORDER IS IMPORTANT !!!
If in the ratio
a
/b
you wrote
…………… then the side adjacent (next) to the “a”
the “a” on the numerator, then…..
(in this case the “f”) must also be the numerator.
If you would have written
the ratio
……………then the side adjacent (next) to the “b”
b
/a
(in this case the “g”) must also be the numerator
b
a
12
=
g
f